| L(s) = 1 | − 2.23·2-s + 3.00·4-s + 5-s + 7-s − 2.23·8-s − 2.23·10-s + 2.47·11-s − 4.47·13-s − 2.23·14-s − 0.999·16-s + 2·17-s + 6.47·19-s + 3.00·20-s − 5.52·22-s − 4·23-s + 25-s + 10.0·26-s + 3.00·28-s + 2·29-s + 10.4·31-s + 6.70·32-s − 4.47·34-s + 35-s + 10.9·37-s − 14.4·38-s − 2.23·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + 0.447·5-s + 0.377·7-s − 0.790·8-s − 0.707·10-s + 0.745·11-s − 1.24·13-s − 0.597·14-s − 0.249·16-s + 0.485·17-s + 1.48·19-s + 0.670·20-s − 1.17·22-s − 0.834·23-s + 0.200·25-s + 1.96·26-s + 0.566·28-s + 0.371·29-s + 1.88·31-s + 1.18·32-s − 0.766·34-s + 0.169·35-s + 1.79·37-s − 2.34·38-s − 0.353·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7134400616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7134400616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.62966398931597000531890685261, −10.21125968349440310427970103854, −9.847658785994140335906524698581, −8.978317755589890469018592777050, −7.931019640878645555463656269107, −7.23932452918618201643972522127, −6.06427715487538538134474581204, −4.63388650382293388787577376817, −2.63516891990134838988787971033, −1.17993286506386506043984798835,
1.17993286506386506043984798835, 2.63516891990134838988787971033, 4.63388650382293388787577376817, 6.06427715487538538134474581204, 7.23932452918618201643972522127, 7.931019640878645555463656269107, 8.978317755589890469018592777050, 9.847658785994140335906524698581, 10.21125968349440310427970103854, 11.62966398931597000531890685261
